Please place your completed assignment in the box marked for your tutorial group in the basement of BSB.
(a) Generate 100 samples of size n = 5 from a chi-square distribution on 2 degrees of freedom. What are the true mean and variance of this distribution? Plot histograms of the first three samples.
(i) For each of the 100 samples, compute a "95%" confidence interval for the mean using the formula [x_bar - z0.975 * s / sqrt(n), x_bar + z0.975 * s / sqrt(n)]. Count how many of these confidence intervals include the true mean. Hence give a 99% confidence interval for the actual confidence level.
(ii) For each of the 100 samples, compute a "95%" confidence interval for the mean using the formula [x_bar - tn-1, 0.975 * s / sqrt(n), x_bar + tn-1, 0.975 * s / sqrt(n)] and count how many of these confidence intervals include the true mean. Hence give a 99% confidence interval for the actual confidence level.
(iii) For each of the 100 samples, compute a "95%" confidence for the variance using the formula [(n-1) * s2 / chisqn-1, 0.975, (n-1) * s2 / chisqn-1, 0.025] and count how many of these intervals include the true variance. Hence give a 99% confidence interval for the actual confidence level.
(b) Repeat the above using 100 samples of size n = 50.
(c) Summarize your results and state your conclusions. Do the intervals that miss the true value tend to be more on one side than the other? Are the sample mean and sample variance correlated?
[Hint: To do (a) in Minitab, fill C1-C5 with 100 rows of chisq(2) random numbers, so the 100 samples are in the rows. Use the row statistics option (rmean, rstdev) to put the 100 sample means in C6 and the 100 sample standard deviations in C7. The rest is easy.]
Consider X1, ..., Xn where E[Xi] = m, Var[Xi] = s2, and rij = 1, for all i and j. The data are perfectly correlated. Find E[X_bar], Var[X_bar], E[s2] and Var[s2]. Do these results make sense to you? How could data like these arise in practice?
Consider a test of the hypothesis m = 3 against the alternative m <> 3, where s2 = 3, given n independent normal observations. Plot the power curves for 5% tests with n = 5 and n = 50, putting both curves on the same graph. [Note: the notation <> means "not equal to".]
Graph the probability mass function for a Poisson distribution with mean m = 3. If X follows a Poisson distribution and we wish to test the hypothesis that m = 3 against the alternative that m < 3, with the Type I error rate as close as possible to 5%, what is the rejection region for the test? What is the exact Type I error rate? Draw a graph of the Power Curve for this test. [Note: Use the Poisson distribution, not an approximation.]
Re-draw Figure 8.5 on page 265.
Do 8.37 - 8.42 on page 290.
Do 8.43 - 8.45 on page 290.